An automobile shop manager times six employees and found that the average time it took then to change a water pump was 18 minutes. The SD of the sample was 3. Find the 99% of the interval of the true mean.
99% = mean ± 2.575 SEm
SEm = SD/√(n-1)
To find the 99% confidence interval of the true mean, we can use the formula:
CI = X̄ ± Z * (SD/√n)
Where:
CI = Confidence interval
X̄ = Sample mean
Z = Z-score (critical value)
SD = Standard deviation of the sample
n = Sample size
First, let's calculate the critical value (Z-score) for a 99% confidence level. This critical value represents the number of standard deviations away from the mean that corresponds to a 99% confidence level. We can use a normal distribution table or a calculator to find this value.
The Z-score for a 99% confidence level is approximately 2.576.
Next, let's plug in the given values into the formula:
X̄ = 18 (sample mean)
SD = 3 (standard deviation of the sample)
n = 6 (sample size)
Z = 2.576 (Z-score for a 99% confidence level)
CI = 18 ± 2.576 * (3/√6)
Now, let's calculate the confidence interval:
CI = 18 ± 1.761
The 99% confidence interval of the true mean is (16.239, 19.761).
This means with 99% confidence, we can say that the true mean time it takes to change a water pump falls within the range of 16.239 minutes to 19.761 minutes.